I am analying $$ \sum_{j=k}^{n}\frac{(-1)^{j}{\frac{-3}{2}\choose n-j}}{j(j-1)}\;\;\mbox{where}\;n\ge k\;\mbox{are integers}. $$ By the telescoping series, we have $$ \sum_{j=k}^{n}(-1)^{j}{\frac{-3}{2}\choose n-j}\left(\frac{1}{j-1}-\frac{1}{j}\right). $$
Does it have a closed form? Maybe I should ask that if it is possible to find a closed form for
$$ \sum_{j=k}^{n}\frac{(-1)^{j}{\frac{-3}{2}\choose n-j}}{j}. $$
Please give me some help. Thanks.